再论欧拉数值积分在凸优化加速和稳定性中的作用
Revisiting the Role of Euler Numerical Integration on Acceleration and Stability in Convex Optimization
摘要：
将最优化方法看作是常微分方程的数值积分器，为研究加速一阶优化器提供了一个发人深省的现代框架。在这些文献中，加速度通常被认为与积分器的质量（准确性，能量守恒，辛性）有关。在这项工作中，我们提出了一个新的常微分方程，质疑这种连接：显式和半隐式（又名辛）欧拉离散这个ODE导致加速算法的凸规划。虽然半隐式方法是众所周知的数值分析享受许多理想的功能，物理系统的整合，我们的研究结果表明，这些属性不一定与加速度。
英文摘要：
Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often supposed to be linked to the quality of the integrator (accuracy, energy preservation, symplecticity). In this work, we propose a novel ordinary differential equation that questions this connection: both the explicit and the semi-implicit (a.k.a symplectic) Euler discretizations on this ODE lead to an accelerated algorithm for convex programming. Although semi-implicit methods are well-known in numerical analysis to enjoy many desirable features for the integration of physical systems, our findings show that these properties do not necessarily relate to acceleration.
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